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Replacing the front fork with a rigid one would make a huge difference, especially on the hills. If you investigate your front derailluer, you might find it has plenty of capacity for a larger big chainring, thus saving you the expense of an entirely new crankset. Even if it isn't spec'd for something larger, there's pretty much always room for a chainring two teeth bigger, and depending on what your rear cassette is, the combination of changing the two might well give you enough higher end.
I'm a big advocate of converted MTB's for commuting, particularly up here, but then again it is hard to resist the call of N+1!
Craigslist is full of bikes for fairly reasonable prices, lots of them with low miles, could fulfill your desire for a bike that isn't too expensive.